A367559 Square array T(n, k) = 2^k - n, read by ascending antidiagonals.
1, 0, 2, -1, 1, 4, -2, 0, 3, 8, -3, -1, 2, 7, 16, -4, -2, 1, 6, 15, 32, -5, -3, 0, 5, 14, 31, 64, -6, -4, -1, 4, 13, 30, 63, 128, -7, -5, -2, 3, 12, 29, 62, 127, 256, -8, -6, -3, 2, 11, 28, 61, 126, 255, 512, -9, -7, -4, 1, 10, 27, 60, 125, 254, 511, 1024
Offset: 0
Examples
This sequence as square array T(n, k): n\k 0 1 2 3 4 5 6 7 8 9 10. ---------------------------------------------------------. 0 : 1 2 4 8 16 32 64 128 256 512 1024. 1 : 0 1 3 7 15 31 63 127 255 511 1023. 2 : -1 0 2 6 14 30 62 126 254 510 1022. 3 : -2 -1 1 5 13 29 61 125 253 509 1021. 4 : -3 -2 0 4 12 28 60 124 252 508 1020. 5 : -4 -3 -1 3 11 27 59 123 251 507 1019. 6 : -5 -4 -2 2 10 26 58 122 250 506 1018. 7 : -6 -5 -3 1 9 25 57 121 249 505 1017. 8 : -7 -6 -4 0 8 24 56 120 248 504 1016. 9 : -8 -7 -5 -1 7 23 55 119 247 503 1015. 10: -9 -8 -6 -2 6 22 54 118 246 502 1014.
Links
- Paolo Xausa, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150, flattened).
Crossrefs
Programs
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Mathematica
Table[2^k-n+k,{n,0,10},{k,0,n}] (* Paolo Xausa, Nov 28 2023 *) -
PARI
T(n, k) = 2^k-n \\ Thomas Scheuerle, Nov 23 2023
Formula
G.f. of row n: 1/(1-2*x) - n/(1-x).
E.g.f. of row n: exp(2*x) - n*exp(x).
T(0, k) = 2^k = A000079(k).
T(1, k) = 2^k - 1 = A000225(k).
T(2, k) = 2^k - 2 = A000918(k).
T(3, k) = 2^k - 3 = A036563(k).
T(5, k) = 2^k - 5 = A168616(k).
T(9, k) = 2^k - 9 = A185346(k).
T(10, k) = 2^k - 10 = A246168(k).
T(n, k) = 3*T(n, k-1) - 2*T(n, k-2) for k > 1.
T(n+1, k) = T(n, k) + 1.
T(n, n) = 2^n - n = A000325(n).
Sum_{k = 0..n} T(n - k, k) = A084634(n).
G.f.: (1 - 2*x - y + 3*x*y)/((1 - x)^2*(1 - y)*(1 - 2*y)). - Stefano Spezia, Nov 27 2023