A008594 -id:A008594 - cp's OEIS frontend

A054735 Sums of twin prime pairs.

8, 12, 24, 36, 60, 84, 120, 144, 204, 216, 276, 300, 360, 384, 396, 456, 480, 540, 564, 624, 696, 840, 864, 924, 1044, 1140, 1200, 1236, 1284, 1320, 1620, 1644, 1656, 1716, 1764, 2040, 2064, 2100, 2124, 2184, 2304, 2460, 2556, 2580, 2604, 2640, 2856, 2904

Offset: 1

Enoch Haga, Apr 22 2000

(p^q)+(q^p) calculated modulo pq, where (p,q) is the n-th twin prime pair. Example: (599^601)+(601^599) == 1200 mod (599*601). - Sam Alexander, Nov 14 2003
El'hakk makes the following claim (without any proof): (q^p)+(p^q) = 2*cosh(q arctanh( sqrt( 1-((2/p)^2) ) )) + 2cosh(p arctanh( sqrt( 1-((2/q)^2) ) )) mod p*q. - Sam Alexander, Nov 14 2003
Also: Numbers N such that N/2-1 and N/2+1 both are prime. - M. F. Hasler, Jan 03 2013
Excluding the first term, all remaining terms have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013
Except for the first term, this sequence is a subsequence of A005101 (Abundant numbers) and of A008594 (Multiples of 12). - Ivan N. Ianakiev, Jul 04 2021

			a(3) = 24 because the twin primes 11 and 13 add to 24.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 144.
El'hakk, Page of the time traveler [Archived copy on web.archive.org, as of Oct 28 2009.]

Cf. A001359, A006512, A014574, A040040, A111046.

a054735 = (+ 2) . (* 2) . a001359  -- Reinhard Zumkeller, Feb 10 2015

ZL:=[]:for p from 1 to 1451 do if (isprime(p) and isprime(p+2)) then ZL:=[op(ZL),p+(p+2)]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007
A054735 := proc(n)
2*A001359(n)+2;
end proc: # R. J. Mathar, Jan 06 2013

Select[Table[Prime[n] + 1, {n, 230}], PrimeQ[ # + 1] &] *2 (* Ray Chandler, Oct 12 2005 *)
Total/@Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Oct 23 2022 *)

is_A054735(n)={!bittest(n,0)&&isprime(n\2-1)&&isprime(n\2+1)} \\ M. F. Hasler, Jan 03 2013

pp=1;forprime(p=1,1482, if( p==pp+2, print1(p+pp,", ")); pp=p) \\ Following a suggestion by R. J. Cano, Jan 05 2013

a(n) = 2*A014574(n) = 4*A040040(n) = A111046(n)/2.

a(n) = 12*A002822(n-1) for all n > 1. - M. F. Hasler, Dec 12 2019

Additional comments from Ray Chandler, Nov 16 2003

Broken link fixed by M. F. Hasler, Jan 03 2013

A121032 Multiples of 12 containing a 12 in their decimal representation.

Original entry on oeis.org

12, 120, 312, 612, 912, 1128, 1200, 1212, 1224, 1236, 1248, 1260, 1272, 1284, 1296, 1512, 1812, 2112, 2124, 2412, 2712, 3012, 3120, 3312, 3612, 3912, 4128, 4212, 4512, 4812, 5112, 5124, 5412, 5712, 6012, 6120, 6312, 6612, 6912, 7128, 7212, 7512, 7812

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Subsequence of A008594.

Cf. A121041, A011531, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

import Data.List (isInfixOf)
a121032 n = a121032_list !! (n-1)
a121032_list = filter ((isInfixOf "12") . show) a008594_list
-- Reinhard Zumkeller, Dec 12 2012

Select[12*Range[700],SequenceCount[IntegerDigits[#],{1,2}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 25 2017 *)

is(n)=if(n%12, return(0)); while(n>11, if(n%100==12, return(1)); n\=10); 0 \\ Charles R Greathouse IV, Feb 12 2017

a(n) ~ 12n. - Charles R Greathouse IV, Nov 02 2022

Data fixed by Reinhard Zumkeller, Dec 12 2012

A008595 Multiples of 13.

Original entry on oeis.org

0, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208, 221, 234, 247, 260, 273, 286, 299, 312, 325, 338, 351, 364, 377, 390, 403, 416, 429, 442, 455, 468, 481, 494, 507, 520, 533, 546, 559, 572, 585, 598, 611, 624, 637, 650, 663, 676

Offset: 0

N. J. A. Sloane

Complement of A113763. - Reinhard Zumkeller, Apr 26 2011

Ivan Panchenko, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 325.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A017533, A017545, A076310, A113763, A252994.

A008595:=n->13*n; seq(A008595(n), n=0..100); # Wesley Ivan Hurt, Jan 30 2014

Range[0, 1000, 13] (* Vladimir Joseph Stephan Orlovsky, May 29 2011 *)

a(n)=13*n \\ Charles R Greathouse IV, Jul 10 2016

(floor(a(n)/10) + 4*(a(n) mod 10)) == 0 modulo 13, see A076310. - Reinhard Zumkeller, Oct 06 2002
From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 13*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: 13*x/(x-1)^2. (End)
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 13*x*exp(x).
a(n) = A252994(n)/2. (End)

A183010 a(n) = 24*n - 1.

Original entry on oeis.org

-1, 23, 47, 71, 95, 119, 143, 167, 191, 215, 239, 263, 287, 311, 335, 359, 383, 407, 431, 455, 479, 503, 527, 551, 575, 599, 623, 647, 671, 695, 719, 743, 767, 791, 815, 839, 863, 887, 911, 935, 959, 983, 1007, 1031, 1055, 1079, 1103, 1127, 1151, 1175, 1199

Offset: 0

Omar E. Pol, Jan 21 2011

a(n) is also the denominator of the finite algebraic formula for the number of partitions of n, if n >= 1. The formula is  p(n) = Tr(n)/(24*n - 1), n >= 1. See theorem 1.1 of the Bruinier-Ono paper in the link. For the numerators see A183011.
It appears that a(n) is also the denominator of the coefficient of the third term in the n-th Bruinier-Ono "partition polynomial" H_n(x). See the Bruinier-Ono paper, chapter 5 "Examples". For the numerators see A183007. - Omar E. Pol, Jul 13 2011
Also exponents in the formula q^(-1) + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 11*q^143 + 15*q^167 + ... in which the coefficients are the partition numbers (see A000041, Example section). - Omar E. Pol, Feb 27 2013

			G.f. = -1 + 23*x + 47*x^2 + 71*x^3 + 95*x^4 + 119*x^5 + 143*x^6 + 167*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
A. Dabholkar, S. Murthy, and D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074 [hep-th], 2012-2014, see p. 46.
H. Gupta, Congruent properties of sigma(n), Math. Student 13 (1945) 25-29.
E. Larson and L. Rolen, Integrality properties of the CM-values of certain weak Maass forms, arXiv:1107.4114 [math.NT], 2011.
K. Ono, Congruences for the Andrews spt-function, (see 2.1 Producing modular forms)
W. Sierpinski, Elementary Theory of numbers, Monografie Mathematyczne, vol. 42 (1964) chapt 4, p. 168.
Leo Tavares, Illustration: Star Pairs
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000041, A000203, A008606, A134517 (subset of primes), A183009, A183011, A187206, A280097 (sum of divisors), A280098.

Cf. A008594.

[24*n-1: n in [0..50]]; // G. C. Greubel, Aug 14 2018

Range[23, 2000, 24] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)
(24*Range[0,50])-1 (* Harvey P. Dale, Mar 28 2015 *)

a(n)=24*n-1 \\ Charles R Greathouse IV, Jun 14 2011

a(n) = A008606(n) - 1.
a(1)=23, a(2)=47, a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jan 23 2011
a(n) = A183011(n)/A000041(n). - Omar E. Pol, Jul 14 2011
24 * A280098(n) = A000203(a(n)) if n>0. - Michael Somos, Dec 25 2016
E.g.f.: (24*x-1)*exp(x). - G. C. Greubel, Aug 14 2018
G.f.: (-1 + 25*x)/(1-x)^2. - Wolfdieter Lang, Dec 10 2021
a(n) = 2*A008594(n) - 1. - Leo Tavares, Jun 06 2023

A017629 a(n) = 12*n + 9.

Original entry on oeis.org

9, 21, 33, 45, 57, 69, 81, 93, 105, 117, 129, 141, 153, 165, 177, 189, 201, 213, 225, 237, 249, 261, 273, 285, 297, 309, 321, 333, 345, 357, 369, 381, 393, 405, 417, 429, 441, 453, 465, 477, 489, 501, 513, 525, 537, 549, 561, 573, 585, 597, 609, 621, 633

Offset: 0

N. J. A. Sloane

Numbers k such that k mod 2 = (k+1) mod 3 = 1 and (k+2) mod 4 != 1. - Klaus Brockhaus, Jun 15 2004
For n > 3, the number of squares on the infinite 3-column chessboard at <= n knight moves from any fixed point. - Ralf Stephan, Sep 15 2004
A016946 is the subsequence of squares (for n = 3*k*(k+1) = A028896(k), then a(n) = (6k+3)^2 = A016946(k)). - Bernard Schott, Apr 05 2021

Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequences include A072065, A016945, A016813, A008585, and A005408.

Cf. A008594, A017533, A017545, A017137, A004767, A001019, A028896, A016946, A089911.

a017629 = (+ 9) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

12*Range[0,200]+9 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)
LinearRecurrence[{2,-1},{9,21},60] (* Harvey P. Dale, Apr 14 2019 *)

a(n)=12*n+9 \\ Charles R Greathouse IV, Jul 10 2016

[i+9 for i in range(525) if gcd(i,12) == 12] # Zerinvary Lajos, May 21 2009

a(n) = 6*(4*n+1) - a(n-1) (with a(0)=9). - Vincenzo Librandi, Dec 17 2010
A089911(2*a(n)) = 4. - Reinhard Zumkeller, Jul 05 2013
G.f.: (9 + 3*x)/(1 - x)^2. - Alejandro J. Becerra Jr., Jul 08 2020
Sum_{n>=0} (-1)^n/a(n) = (Pi + log(3-2*sqrt(2)))/(12*sqrt(2)). - Amiram Eldar, Dec 12 2021
E.g.f.: 3*exp(x)*(3 + 4*x). - Stefano Spezia, Feb 25 2023

A008596 Multiples of 14.

Original entry on oeis.org

0, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196, 210, 224, 238, 252, 266, 280, 294, 308, 322, 336, 350, 364, 378, 392, 406, 420, 434, 448, 462, 476, 490, 504, 518, 532, 546, 560, 574, 588, 602, 616, 630, 644, 658, 672, 686, 700, 714, 728

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 326.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A008594, A008595, A033571, A135628, A158186.

Range[0, 1000, 14] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
CoefficientList[Series[14 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)

a(n)=14*n \\ Charles R Greathouse IV, Sep 28 2015

a(n) = A033571(n) - A158186(n). - Reinhard Zumkeller, Mar 13 2009
From R. J. Mathar, Jun 23 2009: (Start)
a(n) = 14*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: 14*x/(x-1)^2. (End)
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 14*x*exp(x).
a(n) = 2*A008589(n) = A135628(n)/2. (End)

A089911 a(n) = Fibonacci(n) mod 12.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1

Offset: 0

Casey Mongoven, Nov 14 2003

From Reinhard Zumkeller, Jul 05 2013: (Start)
Sequence has been applied by several composers to 12-tone equal temperament pitch structure. The complete Fibonacci mod 12 system (a set of 10 periodic sequences) exhausts all possible ordered dyads; that is, every possible combination of two pitches is found in these sets.
a(A008594(n)) = 0;
a(A227144(n)) = 1;
a(3*A047522(n)) = 2;
a(A017569(n)) = a(2*A016933(n)) = a(4*A016777(n)) = 3;
a(2*A017629(n)) = a(3*A017137(n)) = a(6*A004767(n)) = 4;
a(A227146(n)) = 5;
a(nonexistent) = 6;
a(2*A017581(n)) = 7;
a(2*A017557(n)) = a(4*A016813(n)) = 8;
a(A017617(n)) = a(2*A016957(n)) = a(4*A016789(n)) = 9;
a(3*A047621(n)) = 10;
a(2*A017653(n)) = 11. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..1199
Ron Knott, Fibonacci Numbers and the Golden Section
M. Renault, The Fibonacci Sequence Modulo M
A. P. Shah, Fibonacci Sequence Modulo m, Fibonacci Quarterly, Vol.6, No.2 (1968), 139-141.
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
Index entries for linear recurrences with constant coefficients, signature (1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1).

Cf. A000045, A001175, A003893, A010881.

a089911 n = a089911_list !! n
a089911_list = 0 : 1 : zipWith (\u v -> (u + v) `mod` 12)
                       (tail a089911_list) a089911_list
-- Reinhard Zumkeller, Jul 01 2013

[Fibonacci(n) mod 12: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

with(combinat,fibonacci); A089911 := proc(n) fibonacci(n) mod 12; end;

Table[Mod[Fibonacci[n], 12], {n, 0, 100}] (* Vincenzo Librandi, Feb 04 2014 *)

a(n)=fibonacci(n)%12 \\ Charles R Greathouse IV, Feb 03 2014

Has period of 24, restricted period 12 and multiplier 5.

a(n) = (a(n-1) + a(n-2)) mod 12, a(0) = 0, a(1) = 1.

More terms from Ray Chandler, Nov 15 2003

A017545 a(n) = 12*n + 2.

Original entry on oeis.org

2, 14, 26, 38, 50, 62, 74, 86, 98, 110, 122, 134, 146, 158, 170, 182, 194, 206, 218, 230, 242, 254, 266, 278, 290, 302, 314, 326, 338, 350, 362, 374, 386, 398, 410, 422, 434, 446, 458, 470, 482, 494, 506, 518, 530, 542, 554, 566, 578, 590, 602, 614, 626, 638

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 40 ).

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A017533, A017557.

Subsequence of A072065.

List([0..60], n-> 2*(6*n+1) ); # G. C. Greubel, Sep 18 2019

[12*n+2: n in [0..60]]; // Vincenzo Librandi, Jun 07 2011

A017545:=n->12*n+2: seq(A017545(n), n=0..60); # Wesley Ivan Hurt, Apr 27 2017

12*Range[0,60]+2 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

a(n)=12*n+2 \\ Charles R Greathouse IV, Jul 10 2016

[2*(6*n+1) for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jun 07 2011
From G. C. Greubel, Sep 18 2019: (Start)
G.f.: 2*(1 + 5*x)/(1-x)^2.
E.g.f.: 2*(1 + 6*x)*exp(x). (End)
Sum_{n>=0} (-1)^n/a(n) = Pi/12 + sqrt(3)*log(2 + sqrt(3))/12. - Amiram Eldar, Dec 12 2021

A017569 a(n) = 12*n + 4.

Original entry on oeis.org

4, 16, 28, 40, 52, 64, 76, 88, 100, 112, 124, 136, 148, 160, 172, 184, 196, 208, 220, 232, 244, 256, 268, 280, 292, 304, 316, 328, 340, 352, 364, 376, 388, 400, 412, 424, 436, 448, 460, 472, 484, 496, 508, 520, 532, 544, 556, 568, 580, 592, 604, 616, 628

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(46).
Number of 6 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (11;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1A008574; m=3: A016933; m=4: A022144; m=5: A017293. - Sergey Kitaev, Nov 13 2004
Except for 4, exponents e such that x^e - x^2 + 1 is reducible.
If Y and Z are 2-blocks of a (3n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 28 2007
Terms are perfect squares iff n is a generalized octagonal number (A001082), then n = k*(3*k-2) and a(n) = (2*(3*k-1))^2. - Bernard Schott, Feb 26 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A017533, A017545, A089911.
Cf. A016777, A016933, A017293, A022144.
Cf. A001082.

a017569 = (+ 4) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

[12*n+4: n in [0..50]]; // Vincenzo Librandi, May 04 2011

12*Range[0,200]+4 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

A089911(a(n)) = 3. - Reinhard Zumkeller, Jul 05 2013
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(2)/12. - Amiram Eldar, Dec 12 2021
From Stefano Spezia, Feb 25 2023: (Start)
O.g.f.: 4*(1 + 2*x)/(1 - x)^2.
E.g.f.: 4*exp(x)*(1 + 3*x). (End)
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = 2*A016933(n) = 4*A016777(n) = A016777(4*n+1). (End)

A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

Original entry on oeis.org

1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 1

Reinhard Zumkeller, May 24 2004

(T(n,k) mod 4) <> 2, see A042965, A016825.
All numbers m occur A034178(m) times.
The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).

			n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
   1;
   4,  3;
   9,  8,  5;
  16, 15, 12,  7;
  25, 24, 21, 16,  9;
  36, 35, 32, 27, 20, 11;
  49, 48, 45, 40, 33, 24, 13;
  64, 63, 60, 55, 48, 39, 28, 15;
  81, 80, 77, 72, 65, 56, 45, 32, 17;
  ... etc. - _Philippe Deléham_, Mar 07 2013

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A000290, A000567, A004736, A005408, A005563, A008586, A008590.
Cf. A008594, A008598, A016825, A016945, A017329, A028347, A028560.
Cf. A028566, A034178, A042965, A094727, A120070, A128624.
Cf. A000384 (alternating row sums), A002412 (row sums).

[n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024

Table[n^2 - k^2, {n,12}, {k,0,n-1}]//Flatten (* Michael De Vlieger, Nov 25 2015 *)

flatten([[n^2-k^2 for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 12 2024

Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
T(n, k) = A004736(n,k)*A094727(n,k).
T(n, 0) = A000290(n).
T(n, 1) = A005563(n-1) for n>1.
T(n, 2) = A028347(n) for n>2.
T(n, 3) = A028560(n-3) for n>3.
T(n, 4) = A028566(n-4) for n>4.
T(n, n-1) = A005408(n).
T(n, n-2) = A008586(n-1) for n>1.
T(n, n-3) = A016945(n-2) for n>2.
T(n, n-4) = A008590(n-2) for n>3.
T(n, n-5) = A017329(n-3) for n>4.
T(n, n-6) = A008594(n-3) for n>5.
T(n, n-8) = A008598(n-2) for n>7.
T(A005408(k), k) = A000567(k).
Sum_{k=0..n} T(n, k) = A002412(n) (row sums).
From G. C. Greubel, Mar 12 2024: (Start)
Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
G.f.: x*(1 - 3*x^2*y + x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, Aug 04 2025

cp's OEIS Frontend

A054735 Sums of twin prime pairs.

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A121032 Multiples of 12 containing a 12 in their decimal representation.

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A008595 Multiples of 13.

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A183010 a(n) = 24*n - 1.

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A017629 a(n) = 12*n + 9.

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A008596 Multiples of 14.

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A089911 a(n) = Fibonacci(n) mod 12.

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