A074377 -id:A074377 - cp's OEIS frontend

A104569 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product Q*R of the infinite lower triangular matrices Q = [1; 1,3; 1,3,1; 1 3,1,3; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

1, 4, 3, 5, 4, 1, 8, 7, 4, 3, 9, 8, 5, 4, 1, 12, 11, 8, 7, 4, 3, 13, 12, 9, 8, 5, 4, 1, 16, 15, 12, 11, 8, 7, 4, 3, 17, 16, 13, 12, 9, 8, 5, 4, 1, 20, 19, 16, 15, 12, 11, 8, 7, 4, 3, 21, 20, 17, 16, 13, 12, 9, 8, 5, 4, 1, 24, 23, 20, 19, 16, 15, 12, 11, 8, 7, 4, 3, 25, 24, 21, 20, 17, 16, 13, 12, 9, 8, 5, 4, 1

Offset: 1

Gary W. Adamson, Mar 16 2005

			The first few rows of the triangle are:
  1;
  4, 3;
  5, 4, 1;
  8, 7, 4, 3;
  9, 8, 5, 4, 1;
  ...

Cf. A035608, A074377, A104570.

Row sums yield A074377. Columns 1, 3, 5, ... (starting at the diagonal entry) yield A042948. Columns 2, 4, 6, ... (starting at the diagonal entry) yield A014601. The product R*Q yields A104570.

T:=proc(i,j) if j>i then 0 elif i+j mod 2 = 1 then 2*(i-j)+2 elif i mod 2 = 1 and j mod 2 = 1 then 2*(i-j)+1 elif i mod 2 = 0 and j mod 2 = 0 then 2*(i-j)+3 else fi end: for i from 1 to 13 do seq(T(i,j),j=1..i) od; # yields sequence in triangular form # Emeric Deutsch, Mar 23 2005

Q[i_, j_] := If[j <= i, 2 + (-1)^j, 0];
R[i_, j_] := If[j <= i, 1, 0];
T[i_, j_] := Sum[Q[i, k]*R[k, j], {k, 1, 13}];
Table[T[i, j], {i, 1, 13}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jul 24 2024 *)

For 1<=j<=i: T(i, j)=2(i-j+1) if i and j are of opposite parity; T(i, j)=2(i-j)+1 if both i and j are odd; T(i, j)=2(i-j)+3 if both i and j are even. - Emeric Deutsch, Mar 23 2005

More terms from Emeric Deutsch, Mar 23 2005

A104570 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product R*Q of the infinite lower triangular matrices R = [1; 1,1; 1,1,1; 1,1,1,1; ...] and Q = [1; 1,3; 1,3,1; 1,3,1,3; ...].

Original entry on oeis.org

1, 2, 3, 3, 6, 1, 4, 9, 2, 3, 5, 12, 3, 6, 1, 6, 15, 4, 9, 2, 3, 7, 18, 5, 12, 3, 6, 1, 8, 21, 6, 15, 4, 9, 2, 3, 9, 24, 7, 18, 5, 12, 3, 6, 1, 10, 27, 8, 21, 6, 15, 4, 9, 2, 3, 11, 30, 9, 24, 7, 18, 5, 12, 3, 6, 1, 12, 33, 10, 27, 8, 21, 6, 15, 4, 9, 2, 3, 13, 36, 11, 30, 9, 24, 7, 18, 5, 12, 3, 6, 1

Offset: 1

Gary W. Adamson, Mar 16 2005

			First few rows of the triangle:
  1;
  2, 3;
  3, 6, 1;
  4, 9, 2, 3;
  ...

Cf. A035608, A074377, A104569.

Row sums yield A035608. The product Q*R yields A104569.

T:=proc(i,j) if j>i then 0 elif j mod 2 = 1 then i-j+1 else 3*(i-j+1) fi end:for i from 1 to 14 do seq(T(i,j),j=1..i) od; # yields sequence in triangular form # Emeric Deutsch, Mar 23 2005

Q[i_, j_] := If[j <= i, 2 + (-1)^j, 0];
R[i_, j_] := If[j <= i, 1, 0];
T[i_, j_] := Sum[R[i, k]*Q[k, j], {k, 1, 13}];
Table[T[i, j], {i, 1, 13}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jul 24 2024~ *)

Even columns (offset) = 1, 2, 3, ...; while odd columns = 3, 6, 9, ...

T(i,j) = i-j+1 if j <= i and j is odd; 3(i-j+1) if j <= i and j is even. - Emeric Deutsch, Mar 23 2005

More terms from Emeric Deutsch, Mar 23 2005

A158056 a(n) = 16n^2 + 2n.

Original entry on oeis.org

18, 68, 150, 264, 410, 588, 798, 1040, 1314, 1620, 1958, 2328, 2730, 3164, 3630, 4128, 4658, 5220, 5814, 6440, 7098, 7788, 8510, 9264, 10050, 10868, 11718, 12600, 13514, 14460, 15438, 16448, 17490, 18564, 19670, 20808, 21978, 23180, 24414, 25680

Offset: 1

Vincenzo Librandi, Mar 12 2009

The identity (16*n + 1)^2 - (16*n^2 + 2*n)*4^2 = 1 can be written as A158057(n)^2 - a(n)*4^2 = 1. - Vincenzo Librandi, Feb 09 2012

Sequence found by reading the line from 18, in the direction 18, 68, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A158057.

I:=[18, 68, 150]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]];

LinearRecurrence[{3,-3,1},{18,68,150},50]
Table[16n^2+2n,{n,40}]  (* Harvey P. Dale, Apr 13 2011 *)

PARI
```
a(n) = 16*n^2 + 2*n.
```

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: 2*x*(-9 - 7*x)/(x-1)^3.

A158058 a(n) = 16n^2 - 2n.

Original entry on oeis.org

14, 60, 138, 248, 390, 564, 770, 1008, 1278, 1580, 1914, 2280, 2678, 3108, 3570, 4064, 4590, 5148, 5738, 6360, 7014, 7700, 8418, 9168, 9950, 10764, 11610, 12488, 13398, 14340, 15314, 16320, 17358, 18428, 19530, 20664, 21830, 23028, 24258, 25520

Offset: 1

Vincenzo Librandi, Mar 12 2009

The identity (16*(n-1) + 15)^2 - (16*n^2 - 2*n)*4^2 = 1 can be written as A125169(n-1)^2 - a(n)*4^2 = 1. - Vincenzo Librandi, Feb 01 2012
Sequence found by reading the line from 14, in the direction 14, 60, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
The continued fraction expansion of sqrt(a(n)) is [4n-1; {1, 2, 1, 8n-2}]. - Magus K. Chu, Nov 08 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (row 15 in the first table at p. 85, case d(t) = t*(4^2*t-2)).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A125169.

Magma
```
[16*n^2-2*n: n in [1..40]]
```

seq(16*n^2-2*n,n=1..40); # Nathaniel Johnston, Jun 26 2011

LinearRecurrence[{3,-3,1},{14,60,138},40]

PARI
```
a(n) = 16*n^2-2*n.
```

G.f.: x*(-14 - 18*x)/(x-1)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

A158444 a(n) = 16*n^2 + 4.

Original entry on oeis.org

20, 68, 148, 260, 404, 580, 788, 1028, 1300, 1604, 1940, 2308, 2708, 3140, 3604, 4100, 4628, 5188, 5780, 6404, 7060, 7748, 8468, 9220, 10004, 10820, 11668, 12548, 13460, 14404, 15380, 16388, 17428, 18500, 19604, 20740, 21908, 23108, 24340, 25604, 26900, 28228

Offset: 1

Vincenzo Librandi, Mar 19 2009

The identity (8*n^2 + 1)^2 - (16*n^2 + 4)*(2*n)^2 = 1 can be written as A081585(n)^2 - a(n)*A005843(n)^2 = 1. [rewritten by Bruno Berselli, Sep 06 2011]

Sequence found by reading the line from 20, in the direction 20, 68, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A053755, A074377, A081585.

Magma
```
[16*n^2+4: n in [1..50]];
```

a[n_] := 16*n^2 + 4; Array[a, 50] (* Amiram Eldar, Mar 05 2023 *)

a(n)=16*n^2+4 \\ Charles R Greathouse IV, Jun 17 2017

From Bruno Berselli, Sep 06 2011: (Start)
G.f.: 4*x*(5 + 2*x + x^2)/(1-x)^3.
a(n) = 4*A053755(n). (End)
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/2)*Pi/2 - 1)/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/2)*Pi/2)/8. (End)
E.g.f.: 4*(exp(x)*(4*x^2 + 4*x + 1) - 1). - Elmo R. Oliveira, Jan 27 2025

A253187 Number of ordered ways to write n as the sum of a pentagonal number, a second pentagonal number and a generalized decagonal number.

Original entry on oeis.org

1, 2, 2, 2, 1, 1, 1, 3, 4, 2, 2, 1, 4, 3, 3, 4, 2, 3, 1, 3, 2, 2, 5, 3, 3, 3, 3, 6, 3, 6, 4, 2, 3, 1, 7, 2, 4, 5, 5, 4, 1, 5, 5, 2, 3, 4, 4, 5, 5, 5, 3, 5, 7, 6, 4, 3, 1, 6, 6, 8, 5, 3, 6, 4, 7, 4, 2, 6, 5, 5, 3, 4, 8, 3, 3, 3, 6, 6, 7, 9, 6, 2, 5, 6, 7, 7, 4, 6, 6, 7, 5, 3, 10, 6, 3, 4, 5, 7, 3, 10, 7

Offset: 0

Zhi-Wei Sun, Apr 07 2015

nonn

Conjecture: a(n) > 0 for all n. Also, for any ordered pair (k,m) among (5,7), (5,9), (5,13), (6,5), (6,7), (7,5), each nonnegative integer n can be written as the sum of a k-gonal number, a second k-gonal number and a generalized m-gonal number.

See also the author's similar conjectures in A254574, A254631, A255916 and the two linked papers.

			a(33) = 1 since 33 = 0*(3*0-1)/2 + 4*(3*4+1)/2 + 1*(4*1+3).
a(56) = 1 since 56 = 4*(3*4-1)/2 + 2*(3*2+1)/2 + 3*(4*3+3).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.
Zhi-Wei Sun, On universal sums a*x^2+b*y^2+f(z), a*T_x+b*T_y+f(z) and a*T_x+b*y^2+f(z), arXiv:1502.03056 [math.NT], 2015.

Cf. A000326, A000384, A000566, A005449, A014105, A074377, A085787, A147875, A254574, A254631.

DQ[n_]:=IntegerQ[Sqrt[16n+9]]
Do[r=0;Do[If[DQ[n-x(3x-1)/2-y(3y+1)/2],r=r+1],{x,0,(Sqrt[24n+1]+1)/6},{y,0,(Sqrt[24(n-x(3x-1)/2)+1]-1)/6}];
Print[n," ",r];Continue,{n,0,100}]

A274832 Values of n such that 2n+1 and 7n+1 are both triangular numbers (A000217).

Original entry on oeis.org

0, 27, 297, 24570, 267030, 22064157, 239792967, 19813588740, 215333817660, 17792580624687, 193369528466037, 15977717587380510, 173645621228683890, 14347972600887073617, 155933574493829667507, 12884463417879004727880, 140028176249837812737720

Offset: 1

Colin Barker, Jul 08 2016

Intersection of A074377 and A274830.

			27 is in the sequence because 2*27+1 = 55, 7*27+1 = 190, and 55 and 190 are both triangular numbers.

Colin Barker, Table of n, a(n) for n = 1..650
Index entries for linear recurrences with constant coefficients, signature (1,898,-898,-1,1).

Cf. A000217, A074377, A274830.

Cf. A124174 (2*n+1 and 9*n+1), A274579 (2*n+1 and 5*n+1), A274603 (2*n+1 and 3*n+1), A274680 (2*n+1 and 4*n+1), A274756 (2*n+1 and 7*n+1).

LinearRecurrence[{1, 898, -898, -1, 1}, {0, 27, 297, 24570, 267030}, 20] (* Paolo Xausa, Oct 21 2024 *)

isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(7*n+1, 3)

concat(0, Vec(27*x^2*(1+10*x+x^2)/((1-x)*(1-30*x+x^2)*(1+30*x+x^2)) + O(x^20)))

G.f.: 27*x^2*(1+10*x+x^2) / ((1-x)*(1-30*x+x^2)*(1+30*x+x^2)).

A082041 a(n) = 16n^2 + 4n + 1.

Original entry on oeis.org

1, 21, 73, 157, 273, 421, 601, 813, 1057, 1333, 1641, 1981, 2353, 2757, 3193, 3661, 4161, 4693, 5257, 5853, 6481, 7141, 7833, 8557, 9313, 10101, 10921, 11773, 12657, 13573, 14521, 15501, 16513, 17557, 18633, 19741, 20881, 22053, 23257, 24493

Offset: 0

Paul Barry, Apr 02 2003

Also sequence found by reading the segment (1,21) together with the line from 21, in the direction 21, 73, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Column k=4 of A082039.

Cf. A054569, A082040, A074377.

Table[16n^2+4n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,21,73},50] (* Harvey P. Dale, Sep 28 2024 *)

a(n)=16*n^2+4*n+1 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (-1-18*x-13*x^2)/(x-1)^3 . - R. J. Mathar, Dec 03 2014
From Elmo R. Oliveira, Oct 28 2024: (Start)
E.g.f.: exp(x)*(1 + 20*x + 16*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A379697 For n >= 3, a(n) is the least k >= 0 such that (k + 1)(2n + k) / 2 is a triangular number (A000217).

Original entry on oeis.org

0, 2, 5, 0, 1, 20, 4, 0, 2, 6, 8, 2, 0, 10, 119, 3, 4, 20, 0, 1, 5, 14, 2, 5, 1, 0, 32, 6, 7, 464, 20, 2, 8, 0, 24, 8, 2, 4, 65, 9, 10, 47, 0, 3, 11, 30, 17, 2, 3, 1, 59, 12, 0, 2, 21, 4, 14, 38, 40, 14, 4, 42, 101, 0, 16, 74, 2, 5, 17, 46, 48, 17, 5, 1, 11, 0, 19, 125, 10, 6, 20, 54, 1, 20, 6, 44, 272, 21, 0

Offset: 3

Ctibor O. Zizka, Dec 30 2024

Also for n >= 3, a(n) is the least k >= 0 such that the Sum_{i = 0..k} (n + i) is a triangular number (A000217). For n = 0, 1 the Sum is a triangular number for all n. For n = 2, there is no solution.

			n = 4: the least k >= 0 such that (k + 1)*(8 + k)/2 is a triangular number is k = 2, thus a(4) = 2.
n = 6: the least k >= 0 such that (k + 1)*(12 + k)/2 is a triangular number is k = 0, thus a(6) = 0.

Robert Israel, Table of n, a(n) for n = 3..10000

Cf. A000217, A074377, A379676.

f:= proc(n) local t, alpha, beta;
      t:= n^2-n;
      alpha:= convert(select(type, numtheory:-divisors(t),odd),list);
      beta:= map(s -> (s+t/s - 1)/2 - n, alpha);
      min(select(`>=`,beta,0))
end proc:
map(f, [$3..100]); # Robert Israel, Jan 30 2025

a[n_] := Module[{k = 0}, While[! IntegerQ[Sqrt[4*(k + 1)*(2*n + k) + 1]], k++]; k]; Array[a, 100, 3] (* Amiram Eldar, Dec 30 2024 *)

a(n) = my(k=0); while (!ispolygonal((k + 1)*(2*n + k)/2, 3), k++); k; \\ Michel Marcus, Dec 30 2024

a(n) = 0 for n from A000217.

a(n) = 1 for n from A074377 AND n is not a triangular number.

A104571 Triangle T(n,k) = A042948(n-k+1) read by rows, 0<=k<=n.

Original entry on oeis.org

1, 4, 1, 5, 4, 1, 8, 5, 4, 1, 9, 8, 5, 4, 1, 12, 9, 8, 5, 4, 1, 13, 12, 9, 8, 5, 4, 1, 16, 13, 12, 9, 8, 5, 4, 1

Offset: 0

Gary W. Adamson, Mar 16 2005

			The first few rows are:
1;
4, 1;
5, 4, 1;
8, 5, 4, 1;
9, 8, 5, 4, 1;
...

Cf. A042948, A035608 (row sums), A104570, A104569, A074377.

The triangle is extracted from the product of lower triangular matrices (with the rest of the terms all zeros): G * R (or R * G); G = [1; 3, 1; 1, 3, 1; 3, 1, 3, 1;...]; R = [1; 1, 1; 1, 1, 1;...].

cp's OEIS Frontend

A104569 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product Q*R of the infinite lower triangular matrices Q = [1; 1,3; 1,3,1; 1 3,1,3; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

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A104570 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product R*Q of the infinite lower triangular matrices R = [1; 1,1; 1,1,1; 1,1,1,1; ...] and Q = [1; 1,3; 1,3,1; 1,3,1,3; ...].

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A158056 a(n) = 16*n^2 + 2*n.

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Magma

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A158058 a(n) = 16*n^2 - 2*n.

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Magma

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A158444 a(n) = 16*n^2 + 4.

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Magma

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A253187 Number of ordered ways to write n as the sum of a pentagonal number, a second pentagonal number and a generalized decagonal number.

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Mathematica

A274832 Values of n such that 2*n+1 and 7*n+1 are both triangular numbers (A000217).

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Mathematica

A158056 a(n) = 16n^2 + 2n.

A158058 a(n) = 16n^2 - 2n.

A274832 Values of n such that 2n+1 and 7n+1 are both triangular numbers (A000217).

A082041 a(n) = 16n^2 + 4n + 1.

A379697 For n >= 3, a(n) is the least k >= 0 such that (k + 1)(2n + k) / 2 is a triangular number (A000217).