A048488 -id:A048488 - cp's OEIS frontend

A048483 Array read by antidiagonals: T(k,n) = (k+1)2^n - k.

1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 10, 5, 1, 32, 31, 22, 13, 6, 1, 64, 63, 46, 29, 16, 7, 1, 128, 127, 94, 61, 36, 19, 8, 1, 256, 255, 190, 125, 76, 43, 22, 9, 1, 512, 511, 382, 253, 156, 91, 50, 25, 10, 1, 1024, 1023, 766, 509, 316, 187

Offset: 0

Clark Kimberling

n-th difference of (T(k,n),T(k,n-1),...,T(k,0)) is k+1, for n=1,2,3,...; k=0,1,2,...

			1 2 4 8 16 32 ...
1 3 7 15 31 63 ...
1 4 10 22 46 94 ...
1 5 13 29 61 125 ...
1 6 16 36 76 156 ...

Rows are A000079 (k=0), A000225 (k=1), A033484 (k=2), A036563 (k=3), A048487 (k=4), A048488 (k=5), A048489 (k=6), A048490 (k=7), A048491 (k=8).

Main diagonal is A048493. Cf. A048494.

G.f.: (1-x+kx)/[(1-x)(1-2x)]. E.g.f.: (k+1)*exp(2x) - k*exp(x).

Recurrences: T(k, n) = 2T(k, n-1)+k = T(k-1, n)+2^n-1, T(k, 0) = 1.

Edited by Ralf Stephan, Feb 05 2004

A131114 T(n,k) = 6binomial(n,k) - 5I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 6, 1, 6, 12, 1, 6, 18, 18, 1, 6, 24, 36, 24, 1, 6, 30, 60, 60, 30, 1, 6, 36, 90, 120, 90, 36, 1, 6, 42, 126, 210, 210, 126, 42, 1, 6, 48, 168, 336, 420, 336, 168, 48, 1, 6, 54, 216, 504, 756, 756, 504, 216, 54, 1

Offset: 0

Gary W. Adamson, Jun 15 2007

Row sums give A048488.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  6,  1;
  6, 12,  1;
  6, 18, 18,   1;
  6, 24, 36,  24,  1;
  6, 30, 60,  60, 30,  1;
  6, 36, 90, 120, 90, 36, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007318, A048488, A131110, A131111, A131112, A131113, A131115.

T:= function(n,k)
    if k=n then return 1;
    else return 6*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019

[k eq n select 1 else 6*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

seq(seq(`if`(k=n, 1, 6*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

Table[If[k==n, 1, 6*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==n, 1, 6*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019

def T(n, k):
    if (k==n): return 1
    else: return 6*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

T(n,k) = 6*A007318(n,k) - 5*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 5*x - x*y)/((1 - x*y)*(1 - x - x*y)).

A079360 Sequence of sums of alternating increasing powers of 2.

Original entry on oeis.org

1, 5, 7, 15, 19, 35, 43, 75, 91, 155, 187, 315, 379, 635, 763, 1275, 1531, 2555, 3067, 5115, 6139, 10235, 12283, 20475, 24571, 40955, 49147, 81915, 98299, 163835, 196603, 327675, 393211, 655355, 786427, 1310715, 1572859, 2621435, 3145723

Offset: 0

Cino Hilliard, Feb 15 2003

Found as a question on http://mail.python.org/mailman/listinfo/tutor poster: reavey.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Tutor User "reavey" (reavey@nep.net) and others, How to write an algorithm for sequence, Tutor -- Discussion for learning programming with Python, 2003.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A079361, A079362, A048488 (bisection).

a:=[1,5,7];; for n in [4..30] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Aug 07 2019

I:=[1,5,7]; [n le 3 select I[n] else Self(n-1) +2*Self(n-2) -2*Self(n-3): n in [1..40]]; // G. C. Greubel, Aug 07 2019

seq(coeff(series((1+4*x)/((1-x)*(1-2*x^2)), x, n+1), x, n), n = 0..40); # G. C. Greubel, Aug 07 2019

LinearRecurrence[{1,2,-2}, {1,5,7}, 40] (* G. C. Greubel, Aug 07 2019 *)

seq(n) = { j=a=1; p=2; print1(1" "); while(j<=n, a = a + 2^p; print1(a" "); a = a+2^(p-1); print1(a" "); p+=1; j+=2; ) }

PARI
```
a(n)=if(n<0,0,(6-n%2)*2^ceil(n/2)-5)
```

@CachedFunction
def a(n):
    if (n==0): return 1
    elif (1<=n<=2): return nth_prime(n+2)
    else: return a(n-1) + 2*a(n-2) - 2*a(n-3)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 07 2019

a(2n) = 6*2^n - 5, a(2n-1) = 5*(2^n - 1). - Benoit Cloitre, Feb 16 2003
From Colin Barker, Sep 19 2012: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).
G.f.: (1+4*x)/((1-x)*(1-2*x^2)). (End)

A119727 Triangular array: T(n,k) = T(n,n) = 1, T(n,k) = 5T(n-1, k-1) + 2T(n-1, k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 19, 37, 1, 1, 43, 169, 187, 1, 1, 91, 553, 1219, 937, 1, 1, 187, 1561, 5203, 7969, 4687, 1, 1, 379, 4057, 18211, 41953, 49219, 23437, 1, 1, 763, 10009, 56707, 174961, 308203, 292969, 117187, 1, 1, 1531, 23833, 163459, 633457, 1491211, 2126953, 1699219, 585937, 1

Offset: 1

Zerinvary Lajos, Jun 14 2006

Second column is A048488. Second diagonal is A057651.

			Triangle begins as:
  1;
  1,    1;
  1,    7,     1;
  1,   19,    37,      1;
  1,   43,   169,    187,      1;
  1,   91,   553,   1219,    937,       1;
  1,  187,  1561,   5203,   7969,    4687,       1;
  1,  379,  4057,  18211,  41953,   49219,   23437,       1;
  1,  763, 10009,  56707, 174961,  308203,  292969,  117187,      1;
  1, 1531, 23833, 163459, 633457, 1491211, 2126953, 1699219, 585937, 1;

TERMESZET VILAGA XI.TERMESZET-TUDOMANY DIAKPALYAZAT 133.EVF. 6.SZ. jun. 2002. Vegh Lea (and Vegh Erika): "Pascal-tipusu haromszogek" http://www.kfki.hu/chemonet/TermVil/tv2002/tv0206/tartalom.html

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A007318, A048488, A057651, A119725, A119726.

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return 5*T(n-1,k-1) + 2*T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 18 2019

T:= proc(n, k) option remember;
      if k=1 and k=n then 1
    else 5*T(n-1, k-1) + 2*T(n-1, k)
      fi
end: seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Nov 18 2019

T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 5*T[n-1, k-1] + 2*T[n-1, k]]; Table[T[n,k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==1 || k==n, 1, 5*T(n-1,k-1) + 2*T(n-1,k)); \\ G. C. Greubel, Nov 18 2019

@CachedFunction
def T(n, k):
    if (k==1 or k==n): return 1
    else: return 5*T(n-1, k-1) + 2*T(n-1, k)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 18 2019

Edited by Don Reble, Jul 24 2006

A062001 Table by antidiagonals of n-Stohr sequences: T(n,k) is least positive integer not the sum of at most n distinct terms in the n-th row from T(n,1) through to T(n,k-1).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 7, 4, 2, 1, 6, 10, 8, 4, 2, 1, 7, 13, 15, 8, 4, 2, 1, 8, 16, 22, 16, 8, 4, 2, 1, 9, 19, 29, 31, 16, 8, 4, 2, 1, 10, 22, 36, 46, 32, 16, 8, 4, 2, 1, 11, 25, 43, 61, 63, 32, 16, 8, 4, 2, 1, 12, 28, 50, 76, 94, 64, 32, 16, 8, 4, 2, 1, 13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1

Offset: 1

Henry Bottomley, May 29 2001

			Array begins as:
  1, 2, 3, 4,  5,  6,  7,   8,   9, ... A000027;
  1, 2, 4, 7, 10, 13, 16,  19,  22, ... A033627;
  1, 2, 4, 8, 15, 22, 29,  36,  43, ... A026474;
  1, 2, 4, 8, 16, 31, 46,  61,  76, ... A051039;
  1, 2, 4, 8, 16, 32, 63,  94, 125, ... A051040;
  1, 2, 4, 8, 16, 32, 64, 127, 190, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 255, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;
Antidiagonal triangle begins as:
   1;
   2,  1;
   3,  2,  1;
   4,  4,  2,  1;
   5,  7,  4,  2,   1;
   6, 10,  8,  4,   2,   1;
   7, 13, 15,  8,   4,   2,  1;
   8, 16, 22, 16,   8,   4,  2,  1;
   9, 19, 29, 31,  16,   8,  4,  2,  1;
  10, 22, 36, 46,  32,  16,  8,  4,  2, 1;
  11, 25, 43, 61,  63,  32, 16,  8,  4, 2, 1;
  12, 28, 50, 76,  94,  64, 32, 16,  8, 4, 2, 1;
  13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Rows include A000027, A033627, A026474, A051039, A051040.
Diagonals include A000079, A000225, A033484, A036563, A048487.
Cf. A048488, A048489, A048490, A048491, A139634, A139635, A139697.
A048483 can be seen as half this table.

T[n_, k_]:= If[kG. C. Greubel, May 03 2022 *)

def A062001(n,k):
    if (kA062001(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, May 03 2022

If k <= n+1 then A(n, k) = 2^(k-1), while if k > n+1, A(n, k) = (2^n - 1)*(k - n) + 1 (array).
T(n, k) = A(k, n-k+1) (antidiagonals).
T(2*n-1, n) = A000079(n-1), n >= 1.
T(2*n, n) = A000079(n), n >= 1.
T(2*n+1, n) = A000225(n+1), n >= 1.
T(2*n+2, n) = A033484(n), n >= 1.
T(2*n+3, n) = A036563(n+3), n >= 1.
T(2*n+4, n) = A048487(n), n >= 1.
From G. C. Greubel, May 03 2022: (Start)
T(n, k) = (2^k - 1)*(n-2*k+1) + 1 for k < n/2, otherwise 2^(n-k).
T(2*n+5, n) = A048488(n), n >= 1.
T(2*n+6, n) = A048489(n), n >= 1.
T(2*n+7, n) = A048490(n), n >= 1.
T(2*n+8, n) = A048491(n), n >= 1.
T(2*n+9, n) = A139634(n), n >= 1.
T(2*n+10, n) = A139635(n), n >= 1.
T(2*n+11, n) = A139697(n), n >= 1. (End)

A057912 Numbers k such that 3*2^k - 5 is prime.

Original entry on oeis.org

2, 3, 4, 7, 9, 10, 13, 15, 25, 31, 34, 48, 52, 64, 109, 145, 162, 204, 207, 231, 271, 348, 444, 553, 559, 1504, 1708, 3048, 3970, 4423, 4668, 5737, 5877, 6130, 8584, 10663, 12517, 16591, 18450, 19362, 22291, 34468, 36637, 52212, 59040, 130279, 236511, 392260, 496411, 536868, 565024, 662703, 908005

Offset: 1

Robert G. Wilson v, Nov 16 2000

a(44) > 44233. - Jinyuan Wang, Feb 02 2020

a(54) > 1000000 - Jon Grantham, Jul 30 2023

Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Cf. A057913 (3*2^k + 5 is prime).

Cf. A048488 (3*2^k - 5, but with different offset).

Do[ If[ PrimeQ[ 3*2^n - 5 ], Print[ n ] ], {n, 1, 3000} ]

is(n)=ispseudoprime(3*2^n-5) \\ Charles R Greathouse IV, Jun 13 2017

a(36)-a(41) from Vincenzo Librandi, Oct 10 2013
a(42)-a(43) from Jinyuan Wang, Feb 02 2020
a(44)-a(45) from Michael S. Branicky, May 20 2023
a(46)-a(53) from Jon Grantham, Jul 30 2023

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

G. C. Greubel, Table of n, a(n) for n = 0..985
Index entries for linear recurrences with constant coefficients, signature (33,-393,1991,-3930,3300,-1000).

Cf. A002275, A011557.

Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)

a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020

[3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

More terms from Jinyuan Wang, Feb 27 2020

A256255 Triangle read by rows: T(n,k) = 6*k + 1, n>=0, 0<=k<=(2^n-1).

Original entry on oeis.org

1, 1, 7, 1, 7, 13, 19, 1, 7, 13, 19, 25, 31, 37, 43, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103

Offset: 0

Omar E. Pol, Apr 30 2015

Row n lists the first 2^n terms of A016921, n >= 0.
Row sums give A165665.
Right border gives A048488.
The sum of all terms of the first k rows gives A060867(k).
The product of the terms of the third row is equal to the Hardy-Ramanujan number: 1 * 7 * 13 * 19 = 1729.

			Triangle begins:
1;
1,7;
1,7,13,19;
1,7,13,19,25,31,37,43;
1,7,13,19,25,31,37,43,49,55,61,67,73,79,85,91;
...
Illustration of initial terms in the fourth quadrant of the square grid:
------------------------------------------------------------------------
n   a(n)             Compact diagram
------------------------------------------------------------------------
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
0    1      |_|_  |_ _ _  |_ _ _ _ _ _ _  |
1    1      | |_| |_ _  | |_ _ _ _ _ _  | |
2    7      |_ _ _|_  | | |_ _ _ _ _  | | |
3    1      | | | |_| | | |_ _ _ _  | | | |
4    7      | | |_ _ _| | |_ _ _  | | | | |
5   13      | |_ _ _ _ _| |_ _  | | | | | |
6   19      |_ _ _ _ _ _ _|_  | | | | | | |
7    1      | | | | | | | |_| | | | | | | |
8    7      | | | | | | |_ _ _| | | | | | |
9   13      | | | | | |_ _ _ _ _| | | | | |
10  19      | | | | |_ _ _ _ _ _ _| | | | |
11  25      | | | |_ _ _ _ _ _ _ _ _| | | |
12  31      | | |_ _ _ _ _ _ _ _ _ _ _| | |
13  37      | |_ _ _ _ _ _ _ _ _ _ _ _ _| |
14  43      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
For other diagrams of the same family see A241717 and A256258.

Paolo Xausa, Table of n, a(n) for n = 0..16382 (rows 0..13 of the triangle, flattened)

Cf. A000225, A001235, A016921, A048488, A060867, A165665, A241717, A256258.

With[{rows=7},Array[Range[1,6*2^#,6]&,rows,0]] (* Paolo Xausa, Sep 26 2023 *)

A368826 Square array T(n,k) = 3*2^k - n read by ascending antidiagonals.

Original entry on oeis.org

3, 2, 6, 1, 5, 12, 0, 4, 11, 24, -1, 3, 10, 23, 48, -2, 2, 9, 22, 47, 96, -3, 1, 8, 21, 46, 95, 192, -4, 0, 7, 20, 45, 94, 191, 384, -5, -1, 6, 19, 44, 93, 190, 383, 768, -6, -2, 5, 18, 43, 92, 189, 382, 767, 1536, -7, -3, 4, 17, 42, 91, 188, 381, 766, 1535, 3072

Offset: 0

Paul Curtz, Jan 07 2024

Similar to A367559.

			Table begins:
   3  6 12 24 48 96 ...
   2  5 11 23 47 95 ...
   1  4 10 22 46 94 ...
   0  3  9 21 45 93 ...
  -1  2  8 20 44 92 ...
  -2  1  7 19 43 91 ...
  ...

Cf. A000225, A000918, A007283, A033484, A048488, A083329, A165751, A367559.

Cf. diagonals: A123720, A079583, A095151, A101945 (after -1).

T[n_, k_] := 3*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jan 15 2024 *)

T(0,k) = 3*2^k     =   A007283(k).
T(1,k) = 3*2^k - 1 =   A083329(k+1).
T(2,k) = 3*2^k - 2 =   A033484(k).
T(3,k) = 3*2^k - 3 = 3*A000225(k).
T(4,k) = 3*2^k - 4 =  -A165751(k).
T(5,k) = 3*2^k - 5 =   A048488(k-1)
T(6,k) = 3*2^k - 6 = 3*A000918(k).

cp's OEIS Frontend

A048483 Array read by antidiagonals: T(k,n) = (k+1)2^n - k.

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A131114 T(n,k) = 6*binomial(n,k) - 5*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

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A079360 Sequence of sums of alternating increasing powers of 2.

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A119727 Triangular array: T(n,k) = T(n,n) = 1, T(n,k) = 5*T(n-1, k-1) + 2*T(n-1, k), read by rows.

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A062001 Table by antidiagonals of n-Stohr sequences: T(n,k) is least positive integer not the sum of at most n distinct terms in the n-th row from T(n,1) through to T(n,k-1).

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A057912 Numbers k such that 3*2^k - 5 is prime.

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Extensions

A131114 T(n,k) = 6binomial(n,k) - 5I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

A119727 Triangular array: T(n,k) = T(n,n) = 1, T(n,k) = 5T(n-1, k-1) + 2T(n-1, k), read by rows.

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.